The function you've provided is:

\[
F(x) = \frac{x - 3}{x^2 - x + 2}
\]

### Analysis of the function:
1. **Domain**: 
   The domain of the function is all real numbers except where the denominator equals zero. Let's find out when the denominator is zero:
   \[
   x^2 - x + 2 = 0
   \]
   The discriminant of this quadratic equation is:
   \[
   \Delta = (-1)^2 - 4(1)(2) = 1 - 8 = -7
   \]
   Since the discriminant is negative, the quadratic equation has no real solutions, meaning the denominator never equals zero for real \(x\). Therefore, the domain of the function is all real numbers, \( (-\infty, \infty) \).

2. **Intercepts**:
   - **x-intercept**: Set \(F(x) = 0\), meaning the numerator must equal zero:
     \[
     x - 3 = 0 \quad \Rightarrow \quad x = 3
     \]
     So, the x-intercept is at \( (3, 0) \).
   
   - **y-intercept**: Set \(x = 0\) in the function:
     \[
     F(0) = \frac{0 - 3}{0^2 - 0 + 2} = \frac{-3}{2}
     \]
     The y-intercept is \( \left( 0, -\frac{3}{2} \right) \).

3. **Asymptotes**:
   - **Vertical Asymptotes**: Since the denominator does not equal zero for any real \(x\), there are no vertical asymptotes.
   
   - **Horizontal Asymptote**: The degrees of the numerator and denominator are equal (both are degree 2). The horizontal asymptote is found by looking at the ratio of the leading coefficients. Since the leading coefficients are \(1\) in the numerator and \(1\) in the denominator, the horizontal asymptote is at \( y = 0 \).

4. **Behavior**:
   - As \(x \to \infty\) or \(x \to -\infty\), \(F(x)\) approaches 0, confirming the horizontal asymptote at \(y = 0\).

Would you like more details on any of these steps or a graph of the function?

### Related questions:
1. How do you find the horizontal asymptote of a rational function?
2. What happens to the function's behavior at very large or very small values of \(x\)?
3. How can you determine whether a rational function has vertical asymptotes?
4. How do discriminants help in determining the number of real roots in a quadratic equation?
5. What are the different ways to analyze intercepts of a function?

**Tip**: When analyzing rational functions, always check the domain by solving the denominator for zero and investigate asymptotic behavior by comparing the degrees of the numerator and denominator.

Explore the rational function F(x) = (x - 3) / (x^2 - x + 2) in detail, including its domain, intercepts, and horizontal asymptote. Learn how to calculate the discriminant and analyze function behavior. Suitable for students in Grades 10-12.

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